Minimal semi-flat-cotorsion replacements and cosupport

Abstract

Over a commutative noetherian ring $R$ of finite Krull dimension, we show that every complex of flat cotorsion $R$-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion complex. Consequently, every $R$-complex can be replaced by a minimal semi-flat-cotorsion complex in the derived category over $R$. Furthermore, we describe structure of semi-flat-cotorsion replacements, by which we recover classic theorems for finitistic dimensions. In addition, we improve some results on cosupport and give a cautionary example. We also explain that semi-flat-cotorsion replacements always exist and can be used to describe the derived category over any associative ring.

arXiv comment28 pages. Final version to appear in Journal of Algebra. We have made a number of minor revisions, including modifications to Lemma 1.1, a new Lemma 3.8, and a corrected Proposition A.11

References in corpus (3)

• On purity and applications to coderived and singularity categories
• Minimal complexes of cotorsion flat modules
• Cosupport computations for finitely generated modules over commutative noetherian rings

Cited by in corpus (1)

• Cosupports and minimal pure-injective resolutions of affine rings